How do I get bevelled edges from the inside of a mouth?
The original file is this:



First, I removed 129 doubling vertices (W -> Remove Doubles).

After some basic tweaking (dissolving some edges and adding 2 loopcuts to make vertices adjacent) got this



Now, mouth and teeth form. I decided to drop original vertices of teeth and combine them with loops of mouth and of the hole head. To do that, I deleted top teeth vertices (X -> Faces), selected one vertice from base of the top teeth and one from base of the bottom teeth and pressed J to make edge. 



I did that for all adjacent vertices forming base of teeth.
Then, I added loopcut (Ctrl + R) across the all head. I combined it with the top mouth edge so it to let me add more loopcuts without creating NGons:



Then I added 2 loopcuts across the head including mouth. I scaled them to 0 by Z axis (S -> Z -> 0) and moved slightly by Z to make teeth slightly smaller. Then I began recreating teeth by simply selecting adjacent vertices and pressing F:



After bottom teeth were done, I added adjacent quantity of vertical loopcuts (in this case 6) across the mouth to let it have roundish edges. Finally, I added Subsurf modifirer and set subdivision level to 2.





There are a lot of problems left, the eyes and whole top part of face should be rebuilt also or the eyes would not be seen as they should be. Also, my approach creates a lot of additional geometry - I tend to not creating NGons so Subsurf working as good as possible.

This is what I did. At the same time, I'm not sure what did OP to make his final image.